ALIGNMENT BETWEEN STANDARDS AND PRACTICES IN MATHEMATICS EDUCATION: EXPERIENCES IN ARIZONA
SOURCE:
Journal of Curriculum and Supervision v12 p344-55 Summer '97
The magazine publisher is the copyright holder of this article and it is reproduced with permission. Further reproduction of this article in violation of the copyright is prohibited.
The National Council of Teachers of Mathematics (NCTM) has been the primary impetus for mathematics reform through the development and wide dissemination of two sets of standards. Curriculum and Evaluation Standards for School Mathematics became the model upon which 41 states revised their state guidelines.(FN1) These standards view mathematics as a dynamic system of concepts and relationships that can be applied to solving real-world problems. However, many practicing teachers continue to teach mathematics as a static set of facts, rules, and procedures to be learned, practiced, and applied.
Professional Standards for Teaching Mathematics also challenges teachers' working models of teaching and learning. The standards reject the behaviorist view of teachers as dispensers and students as receptors of knowledge. Instead, the standards call for teachers to provide learning situations in which students construct their own knowledge. Under teacher guidance, students are expected to use directly materials such as fraction bars, counting cubes, and graphing calculators to solve real-world problems. Of course, this model for teaching requires fundamental changes in what goes on inside classrooms.(FN2)
The evidence shows that reform on the order suggested by the NCTM standards is difficult to translate into practice and to sustain once it is begun.(FN3) Reformers have sought to make changes in curriculums by linking them with changes in assessment. In the early 1990s, 42 states had some form of minimum competency testing directly tied to district-level curriculums.(FN4) Frequently this linkage results in measurement-driven instruction and has not brought hoped-for curriculum change.(FN5)
One major reason for the lack or slow pace of change appears to be the failure of reformers to acknowledge the fundamental nature of educational practice. The accumulated literature supports the notion that change in programs is preceded by changes within the people who use them.(FN6) However, helping people change involves complex and resource-intensive processes and often seems to be left to chance.
Because Arizona's reform efforts appear typical of those in other states, a close examination of the state's experiences may assist large numbers of educational reformers. Consequently, this report highlights results of a statewide study of mathematics education in Arizona, explains the results in the context of educational change, and suggests implications for mathematics education reform efforts. RELATED LITERATURE ON CURRICULUM REFORMS
Educational innovations, such as curriculum revisions, take root slowly because educators must change before the programs with which they work change. Moreover, a school curriculum changes only when significant numbers of teachers within a school begin using that curriculum.(FN7)
To initiate educational change successfully requires that matters of relevance, readiness, and resources be considered. Relevance is the interaction of need for change, teachers' understandings of the curricular change, and what the curricular change offers to teachers and students. Readiness to deal with change processes refers to both teachers and the schools in which they teach. At any one time, efforts must focus on a few important changes so that educators are not overwhelmed. Resources in the form of time, funds, and encouragement are absolutely essential for change processes to function.(FN8)
Furthermore, change is a highly personal experience in which individuals transform their subjective realities to incorporate new ideas. For many educators change is a lengthy process that begins with individuals recognizing a need to change, then ending old practices. Thereafter, individuals spend additional time questioning and reflecting about new practices, determining the fit of practices with their beliefs, and becoming increasingly convinced of the merits of the new practices.(FN9) These internal alterations occur before individuals actually engage in new practices with any degree of confidence or consistency.(FN10)
To implement mathematics curriculum reform standards in classrooms, teachers must view mathematics as a dynamic system of concepts and relationships applicable to the solution of real-world problems instead of as static facts, rules, and procedures. They must also accept that learning is an active process in which teachers help students develop their own understandings.(FN11) Only when these values and beliefs become operational are teachers likely to use the practices suggested by the standards. LARGE-SCALE REFORM EFFORTS
Kentucky and California currently lead state mathematics education reform efforts. Kentucky's efforts to modify K-4 mathematics education were supported by the State Department of Education; mathematics educators; mathematicians; and leaders in education, business, and public policy.(FN12) These individuals implemented statewide plans for professional development of both preservice and inservice teachers.(FN13) In California schools, the "vision" of change shared among district and site (school) leaders specifically required changes in teachers' conceptions of mathematics curriculum and instruction. Then extensive professional development gave teachers opportunities to clarify the vision for themselves.(FN14)
Despite these optimistic reports, Cuban holds little hope that state initiatives can influence teachers' work in classrooms. He says,
the notion that a state can mandate changes in what happens between teachers and students in classrooms is defective at its core because it does not take into account the nature of a teacher's teaching and learning relationships with students.... Reforming a relationship between teachers and children by remote control simply cannot be done.(FN15)
Apparently, studies of the state efforts in Kentucky and California have produced no classroom-based results. REFORM EFFORTS IN ARIZONA In Arizona a committee appointed by the State Board of Education first formulated the Arizona Essential Skills for Mathematics in 1987, patterned on the 1986 California frameworks. Following publication of the NCTM standards in 1989, a second committee reformatted the skills and highlighted their congruence with the NCTM Standards.(FN16)
From the beginning, efforts by the Arizona Department of Education to inform teachers in the state about the essential skills focused on assessment, rather than on either curriculum or instruction. Leaders expected that state performance-based assessment of the Essential Skills would prompt teachers to base classroom instruction on the skills. The department issued several sample assessments that described the anticipated state assessment. In 1993-94 it administered preliminary versions of the mathematics assessments on a statewide basis to students in grades 3, 8, and 12.
The State Department of Education asked district officials to produce local assessment plans that linked their objectives, instructional materials, teaching strategies, and assessment. Of course, the department expected local district objectives to reflect the Essential Skills.(FN17) Some districts planned and offered professional development for teachers to help them understand the Essential Skills and their implications. However, across the state such efforts were sporadic and failed to reach significant numbers of teachers.
Since then, political processes have complicated the Arizona situation. The state superintendent of schools who was in office when the reform efforts were initiated did not stand for reelection in November 1994. Subsequently, the new state superintendent suspended plans for assessment. Revised standards and plans for assessment procedures have been adopted. The emphasis on assessment, rather than on curriculum and instruction, continues to be prominent. METHOD SELECTION
Participants in this study were K-12 mathematics teachers, elementary and middle school principals, and high school mathematics department chairs from 135 public and 42 private schools. Participating schools were chosen using a two-step procedure that employed stratified sampling by urban-rural location and high-low socioeconomic status (SES) levels.
Superintendents, then principals, of all selected schools were contacted to participate in the survey. Several declined for various reasons (e.g., involvement in other projects, curriculum revision). In particular, a number of urban districts chose not to participate because of pending lawsuits concerned with access to personnel files. Because of low participation rates by some SES and location groups, the investigators determined that generalizations for such groups would be limited or misleading. Thus, data were collapsed across SES and location.
Returns ranged from 35 to 45 percent for the first round and 50 to 60 percent for the second round. Responses for the two rounds were compared on 25 key variables. Significant between-round differences were found on only two items, suggesting that nonresponse error was not an important variable. INSTRUMENT DEVELOPMENT
Questionnaire and interview instruments were used to gather information from teachers, principals, and department chairs. Investigators actively chose to use open-ended questions to avoid situations in which respondents could check every box that might appear on a closed-ended questionnaire. The investigators administered early versions of questionnaires and interview protocols to several groups in a pilot study. Based on the results, the instruments were revised. PROCEDURES
Each participant received a personalized cover letter that outlined the study and its purposes, along with a stamped, self-addressed envelope. After approximately five weeks, second-round questionnaires were sent to individuals who had not responded initially. A request on the final page of the questionnaire solicited participation of interviewees. Teachers and administrators who were willing to provide additional information were asked to provide telephone numbers where they could be reached for interviews. From the pool of potential interviewees, the investigators selected teachers from urban and rural districts and, in elementary schools, a range of grade levels. Telephone interviews, conducted by trained interviewers, used detailed follow-up questions and actively sought fully developed responses.
Coding schemes were developed for responses to the open-ended questionnaire items. Investigators developed and critiqued the coding schemes; then they trained student assistants to code the questionnaire responses. High levels of interrater agreement, ranging between 93 and 96 percent, were attained. Subsequently, data were entered into computer files. Data were analyzed through frequency, cross-tabulation, and descriptive statistics from the SPSS procedures. RESULTS DESCRIPTION OF PARTICIPANTS
Two hundred eighty elementary teachers, most of them white females, responded to the questionnaire. Their undergraduate preparation for teaching averaged 9.5 credits in mathematics and 5 credits in mathematics methods; 10 percent or fewer held minors in mathematics. Approximately 50 percent of the teachers had a master's degree or 40 hours beyond their bachelor's degree. They averaged 12.97 years of teaching experience.
One hundred eighty-eight secondary teachers responded. Most were white; 45 percent were female. These teachers averaged 26.7 credits in mathematics and 7 credits in mathematics methods. Approximately 65 percent had a master's degree or 40 hours beyond their bachelor's degree. These teachers had an average of 12.64 years of teaching experience.
More teachers and administrators volunteered than could be interviewed, given time constraints and project staff. Thirty-seven teachers in grades K-8, 27 secondary mathematics teachers, 4 high school mathematics department chairs, and 16 elementary or middle school principals, generally representative of the sample, participated in interviews. CURRICULUMS AND INSTRUCTIONAL PRACTICES In this report, curriculum refers to what is taught to students, a deliberately open definition that acknowledges several levels based on their remoteness from students.(FN18) For example, an institutional curriculum, manifested in guides or scope-sequence charts, displays the district- or school-explicit curriculum. On the other hand, teachers prepare instructional curriculums that incorporate their own values and priorities concerning topics, time spent, method of delivery, and other considerations. Therefore, instructional curriculums differ in the degrees to which they reflect institutional curriculums.
Teachers' primary sources of instructional curriculum serve as important indices of their perspectives on mathematics content. Teachers who rely heavily on district guides, typically derived from Arizona Essential Skills for Mathematics or newly copyrighted texts, may embrace the reform version of content and teaching to a greater degree than teachers using other sources. For this reason, all teachers and department chairs were asked to respond to the open-ended questionnaire item, "What do you use to guide your mathematics curriculum?"
Participants typically listed between one and five sources; most listed more than one. Answers were classified according to categories that emerged in the analysis: textbooks, district or diocese guides, state guidelines, personal beliefs, specific mathematics programs, and other sources. The personal beliefs category encompasses related ideas including educators' experiences and perceptions of the needs of students. Particular programs include designated curriculum projects.
Table 1 reveals that results differed according to participant group. For example, whereas 44.3 percent of the elementary mathematics teachers named textbooks as a curriculum source, 57.1 percent of the mathematics chairs gave this response. Regardless of grade level, teachers use textbooks, district-diocese guides, and state guidelines as major curriculum sources. High school chairs also cited textbooks and state guidelines, but few named district guides as major sources. TEACHERS' KNOWLEDGE OF CONTENT AND TEACHING STRATEGIES
Interviews provided additional information about teachers' evaluation of important content for students. These sessions also provided opportunities to question teachers about their instructional strategies.
Elementary school. Elementary teachers identified two or three topics that they believed to be essential for students at their grade levels. As expected, teachers differed in their choices of topics within and across grade levels. Despite this difference, half of the essential topics mentioned overall were clearly content, mostly arithmetic. About 12 percent of topics pertained to geometry and measurement. Teachers in kindergarten through 2nd grade named process skills, including problem solving, estimation, number sense, and patterns, much more often than did teachers in the other groups. Asked why these topics are important, most teachers indicated that the selected topics provide the foundation for future mathematics learning or for use in life outside of school.
All elementary teacher interviewees acknowledged that they taught fractions. To provide a focus for further inquiry, teachers in grades K-3 were asked, "What is the major idea you want students to understand about fractions as parts of wholes?" In grades 4-5, a similar question was applied to equivalent fractions/common denominators; and in grades 6-8, to the addition or subtraction of fractions with unlike denominators.
Analysis of interviewees' responses revealed that less than half (48 percent) of all teachers' responses incorporated the major mathematics concept. For example, many K-3 teachers' responses omitted any reference to "same-size parts," the major idea in fractional names for parts of wholes. This inadequacy in content understanding was substantiated in teachers' descriptions of how they routinely teach these topics.
Asked how they would teach the particular fraction topic, teacher interviewees' responses were vague. Teachers named manipulative materials suitable for instruction on the topics but provided few details about how these materials were to be used. Their expectation that students should experience actual hands-on use of materials or observe their use by others was not stated. How teachers help students to think about the mathematics ideas was also not communicated clearly. Teachers said that the materials were used to increase student understanding, but they offered no details even when interviewers pressed for answers.
Secondary school. High school teachers were interviewed about essential topics in Algebra I because this course furnishes basic understandings useful in both subsequent courses and the work force. Asked to name essential ideas, 64 percent of all the topics mentioned by the teachers related to solving equations, and 36 percent related to graphing. However, when these teachers were asked expressly to name the topics they teach in Algebra I, every teacher acknowledged teaching the same six equations-inequalities topics, and 64 percent listed graphing. Teachers' reasons for the importance of these topics ranged from preparation for subsequent mathematics study to recognition of the goal to increase students' critical thinking abilities as preparation for work outside of school.
Most high school teachers reported that they taught algebraic equations. They were asked, "What is the major idea you would want your students to know about the addition and multiplication properties of equality?" Analysis of the interviewees' responses revealed that about half (56 percent) of the teachers' responses incorporated the appropriate major idea of balancing equations as the underlying use of the properties. However, other teachers mentioned steps they taught for the solution of equations, an answer that may indicate a view of mathematics as facts and rules. PROFESSIONAL DEVELOPMENT
During the 1993-94 school year only about 49 percent of the elementary teachers and 59 percent of the secondary teachers reported participation in any professional development related to mathematics content or methods. Those who participated actually spent little time in these programs--on average, one workday for elementary teachers and two workdays for secondary teachers. Still, both teachers and their supervisors rated the professional development programs as somewhat useful (ratings of 4.6 to 5.0 on a 7-point scale, where 7 is high).
Despite the brevity of professional development, at least 40 percent of the respondents reported making changes in their teaching strategies. These changes included the use of cooperative learning, improved questioning techniques, and hands-on learning activities, approaches recommended by the standards. Only about 10 percent of the elementary teachers and 6 percent of the secondary teachers reported changes in their knowledge of mathematics content, assessment, or students' mathematics needs. The extent to which this knowledge was applied in classrooms could not be determined.
Principals and high school chairs believed that teachers needed assistance with instruction and assessment to a greater extent than did teachers. Whereas 37.5 percent of principals and 51.7 percent of chairs indicated these teacher needs, only 18.2 percent of elementary teachers and 26.6 percent of secondary teachers viewed themselves as having these needs.
Elementary and middle school principals also indicated that their teachers had a fairly strong need for mathematics content (18.8 percent). However, only 4.8 percent of teachers saw mathematics content as a pressing need. Less than 10 percent of high school chairs and teachers perceived needs in content knowledge. DISCUSSION
The results raise serious doubts that mathematics teachers in Arizona view mathematics content as dynamic or as process-oriented. Although less than half the teachers mention their use of state guidelines, district guidelines, or textbooks, a number of teachers are unaware of these curriculum sources.
Few of the elementary teachers interviewed reported knowledge of emphases on mathematical processes related to problem solving or reasoning. Only about half the teachers at elementary or secondary levels showed satisfactory understanding of the meaning of the mathematics concept targeted in their interviews. Teachers who report using the guidelines may use them to select mathematics topics, but they apparently attend little to what students are to understand or demonstrate about the topics.
On the surface, teachers' instructional strategies include activities similar to those included in the guidelines. Some use small groups and a variety of instructional materials, including manipulatives. However, the teachers seemed unable to articulate how these materials should be used and offered no hypotheses about how students learn with the materials.
Taken together, these results suggest that few Arizona teachers possess the perspectives on mathematics content and teaching described in the national standards and state guidelines. Teachers seem to be unaware of their deficiencies in content and instruction. Also, no teachers indicated a need for additional instruction about either.
Teachers in this study were experienced teachers, averaging more than 12 years of teaching experience. Many teachers completed their preparation for teaching before the initiation of the current wave of mathematics education reforms. Therefore, many have had little opportunity to learn about the reform efforts except, perhaps, in postbaccalaureate course work or other professional development programs.
Although half the elementary teachers and 65 percent of the secondary teachers had completed advanced college coursework, they may have had very little contact with knowledge of mathematics education reform. Only half the teachers in this study had any professional development in mathematics education during the 1993-94 school year. One day of professional development for elementary teachers and two days for secondary teachers, on average, appears to be insufficient.
As shown in this survey, teachers modify their teaching based on the professional development programs in which they are involved, regardless of the meagerness of such programs. Furthermore, teachers indicated their willingness to participate in additional professional development. IMPLICATIONS FOR MATHEMATICS EDUCATION REFORM EFFORTS In recent years many states have initiated efforts to reform mathematics offerings in schools through the use of the NCTM standards as the basis of state guidelines. Typically, these guidelines have been coupled with competency examinations to assess students' performance. In Arizona, this process has produced little reform in classrooms. The process simply ignores the fundamental idea that change resides within teachers, rather than in mandated curriculum or evaluation. The literature of educational innovation is clear that teachers are unlikely to change their behaviors without making changes in their beliefs and values.(FN19)
Results of this study suggest that state-district-diocese guidelines and updated textbooks provide insufficient incentive for teachers to change their practices. Assessment of student performance on new curriculums also is an anemic incentive, largely because both approaches are external to teachers. Missing in the process are changes in teachers' values and beliefs. Teachers who view mathematics as transmission of facts and rules teach this content to their students and are likely to continue to do so until they understand reasons to change.
Large-scale professional development initiatives should be undertaken to help Arizona teachers change their perspectives on mathematics content and teaching procedures. These initiatives must provide teachers with an understanding of the clear need for change, as well as time, resources, and encouragement to implement change in classrooms.
Added material
EVELYN SOWELL is Professor of Education at Arizona State University West, College of Education, P.O. Box 37100, Phoenix, AZ 85069-7100. RON ZAMBO is Assistant Professor of Mathematics Education at Arizona State University West, College of Education, P.O. Box 37100, Phoenix, AZ 85069-7100.
AUTHORS' NOTE: This article is based on a project supported, in part, by the Eisenhower Mathematics and Science Program. Opinions expressed are those of the authors and do not necessarily reflect those of the funding agency. The authors thank Saundra Bryn, Nancy Haas, and three anonymous reviewers for their comments on an earlier draft of this article.
Table 1. Teachers' Sources of Mathematics Curriculum Content (Percentages)
Elementary Middle/Secondary High School
Source (n = 280) (n = 188) (n = 21)
Textbooks 44.3 44.3 57.1
District/diocese guides 43.6 34.6 14.3
State guidelines 38.2 46.8 52.4
Personal beliefs 17.1 20.7 19.0
Specific programs 9.6 0.5 19.0
Other 6.1 10.1 14.3
FOOTNOTES
1 National Council of Teachers of Mathematics, Curriculum and Evaluation Standards for School Mathematics (Reston, VA: NCTM, 1989); Diane Massell, "Setting Standards in Mathematics and Social Studies," Education and Urban Society 26 (February 1994): 118.
2 National Council of Teachers of Mathematics, Curriculum and Evaluation Standards for School Mathematics (Reston, VA: NCTM, 1989); National Council of Teachers of Mathematics, Professional Standards for Teaching Mathematics (Reston, VA: NCTM, 1991).
3 Larry Cuban, "Curriculum Stability and Change," in Handbook of Research on Curriculum, ed. Philip W. Jackson (New York: Macmillan, 1992), pp. 216-247.
4 Chris Pipho, "Centralizing Curriculum at the State Level," in The Politics of Curriculum Decision-Making: Issues in Centralizing the Curriculum, ed. M. Frances Klein (Albany, NY: State University of New York Press, 1991), pp. 67-97.
5 George F. Madaus and Thomas Kellaghan, "Curriculum Evaluation and Assessment," in Handbook of Research on Curriculum, ed. Philip W. Jackson (New York: Macmillan, 1992), pp. 137-147.
6 Since the early 1970s, researchers looking at the Concerns-Based Adoption Model have studied the change processes experienced by teachers and schools as they work on innovations. Among other findings, this research shows that change is a process and that individuals must change before organizations change. See Shirley Hord, Evaluating Educational Innovation (London: Croom Helm, 1987), and Gene E. Hall and Shirley Hord, Change in Schools: Facilitating the Process (Albany, NY: State University of New York Press, 1987). See also George Sharp, Curriculum Development as Re-education of the Teacher (New York: Teachers College, Columbia University, 1951).
7 Michael G. Fullan with Suzanne Stiegelbauer, The New Meaning of Educational Change (New York: Teachers College Press, 1991); Gene E. Hall and Shirley Hord, Change in Schools: Facilitating the Process (Albany, NY: State University of New York Press, 1987); Shirley Hord, Evaluating Educational Innovation (London: Croom Helm, 1987).
8 Michael G. Fullan with Suzanne Stiegelbauer, The New Meaning of Educational Change (New York: Teachers College Press, 1991).
9 Walter Doyle and Gerald A. Ponder, "The Practicality Ethic in Teacher Decision-Making," Interchange 8 (Summer 1977-78): 1-12.
10 William Bridges, Managing Transitions: Making the Most of Change (Reading, MA: Addison Wesley, 1991).
11 Michael T. Battista, "Teacher Beliefs and the Reform Movement in Mathematics Education," Phi Delta Kappan 75 (February 1994): 462-470.
12 William S. Bush, "Implementing the K-4 Mathematics Standards in Kentucky," Arithmetic Teacher 41 (November 1993): 166-169.
13 Ibid.; William S. Bush, "The Kentucky K-4 Mathematics Specialist Program," in Professional Development for Teachers of Mathematics, 1994 NCTM Yearbook, ed. Douglas B. Aichele and Arthur F. Coxford (Reston, VA: NCTM, 1994), pp. 246-254.
14 David D. Marsh and Allan R. Odden, "State-Initiated Curriculum Reform in Elementary School Mathematics and Science Programs" (paper presented at the annual meeting of the American Educational Research Association, Boston, April 1990).
15 Larry Cuban, "State-Powered Curricular Reform, Measurement-Driven Instruction," Phi Kappa Phi Journal 67 (Summer 1987); 23.
16 Arizona Essential Skills for Mathematics (Phoenix, AZ: Arizona Department of Education, 1992).
17 Ibid.
18 John I. Goodlad and Zhixin Su, "Organization of the Curriculum," in Handbook of Research on Curriculum, ed. Philip W. Jackson (New York: Macmillan, 1992), pp. 327-328.
19 Michael G. Fullan with Suzanne Stiegelbauer, The New Meaning of Educational Change (New York: Teachers College Press, 1991).
WBN: 9719602453005
Type in the content of your page here.
The National Council of Teachers of Mathematics (NCTM) has been the primary impetus for mathematics reform through the development and wide dissemination of two sets of standards. Curriculum and Evaluation Standards for School Mathematics became the model upon which 41 states revised their state guidelines.(FN1) These standards view mathematics as a dynamic system of concepts and relationships that can be applied to solving real-world problems. However, many practicing teachers continue to teach mathematics as a static set of facts, rules, and procedures to be learned, practiced, and applied.
Professional Standards for Teaching Mathematics also challenges teachers' working models of teaching and learning. The standards reject the behaviorist view of teachers as dispensers and students as receptors of knowledge. Instead, the standards call for teachers to provide learning situations in which students construct their own knowledge. Under teacher guidance, students are expected to use directly materials such as fraction bars, counting cubes, and graphing calculators to solve real-world problems. Of course, this model for teaching requires fundamental changes in what goes on inside classrooms.(FN2)
The evidence shows that reform on the order suggested by the NCTM standards is difficult to translate into practice and to sustain once it is begun.(FN3) Reformers have sought to make changes in curriculums by linking them with changes in assessment. In the early 1990s, 42 states had some form of minimum competency testing directly tied to district-level curriculums.(FN4) Frequently this linkage results in measurement-driven instruction and has not brought hoped-for curriculum change.(FN5)
One major reason for the lack or slow pace of change appears to be the failure of reformers to acknowledge the fundamental nature of educational practice. The accumulated literature supports the notion that change in programs is preceded by changes within the people who use them.(FN6) However, helping people change involves complex and resource-intensive processes and often seems to be left to chance.
Because Arizona's reform efforts appear typical of those in other states, a close examination of the state's experiences may assist large numbers of educational reformers. Consequently, this report highlights results of a statewide study of mathematics education in Arizona, explains the results in the context of educational change, and suggests implications for mathematics education reform efforts.
RELATED LITERATURE ON CURRICULUM REFORMS
Educational innovations, such as curriculum revisions, take root slowly because educators must change before the programs with which they work change. Moreover, a school curriculum changes only when significant numbers of teachers within a school begin using that curriculum.(FN7)
To initiate educational change successfully requires that matters of relevance, readiness, and resources be considered. Relevance is the interaction of need for change, teachers' understandings of the curricular change, and what the curricular change offers to teachers and students. Readiness to deal with change processes refers to both teachers and the schools in which they teach. At any one time, efforts must focus on a few important changes so that educators are not overwhelmed. Resources in the form of time, funds, and encouragement are absolutely essential for change processes to function.(FN8)
Furthermore, change is a highly personal experience in which individuals transform their subjective realities to incorporate new ideas. For many educators change is a lengthy process that begins with individuals recognizing a need to change, then ending old practices. Thereafter, individuals spend additional time questioning and reflecting about new practices, determining the fit of practices with their beliefs, and becoming increasingly convinced of the merits of the new practices.(FN9) These internal alterations occur before individuals actually engage in new practices with any degree of confidence or consistency.(FN10)
To implement mathematics curriculum reform standards in classrooms, teachers must view mathematics as a dynamic system of concepts and relationships applicable to the solution of real-world problems instead of as static facts, rules, and procedures. They must also accept that learning is an active process in which teachers help students develop their own understandings.(FN11) Only when these values and beliefs become operational are teachers likely to use the practices suggested by the standards.
LARGE-SCALE REFORM EFFORTS
Kentucky and California currently lead state mathematics education reform efforts. Kentucky's efforts to modify K-4 mathematics education were supported by the State Department of Education; mathematics educators; mathematicians; and leaders in education, business, and public policy.(FN12) These individuals implemented statewide plans for professional development of both preservice and inservice teachers.(FN13) In California schools, the "vision" of change shared among district and site (school) leaders specifically required changes in teachers' conceptions of mathematics curriculum and instruction. Then extensive professional development gave teachers opportunities to clarify the vision for themselves.(FN14)
Despite these optimistic reports, Cuban holds little hope that state initiatives can influence teachers' work in classrooms. He says,
the notion that a state can mandate changes in what happens between teachers and students in classrooms is defective at its core because it does not take into account the nature of a teacher's teaching and learning relationships with students.... Reforming a relationship between teachers and children by remote control simply cannot be done.(FN15)
Apparently, studies of the state efforts in Kentucky and California have produced no classroom-based results.
REFORM EFFORTS IN ARIZONA
In Arizona a committee appointed by the State Board of Education first formulated the Arizona Essential Skills for Mathematics in 1987, patterned on the 1986 California frameworks. Following publication of the NCTM standards in 1989, a second committee reformatted the skills and highlighted their congruence with the NCTM Standards.(FN16)
From the beginning, efforts by the Arizona Department of Education to inform teachers in the state about the essential skills focused on assessment, rather than on either curriculum or instruction. Leaders expected that state performance-based assessment of the Essential Skills would prompt teachers to base classroom instruction on the skills. The department issued several sample assessments that described the anticipated state assessment. In 1993-94 it administered preliminary versions of the mathematics assessments on a statewide basis to students in grades 3, 8, and 12.
The State Department of Education asked district officials to produce local assessment plans that linked their objectives, instructional materials, teaching strategies, and assessment. Of course, the department expected local district objectives to reflect the Essential Skills.(FN17) Some districts planned and offered professional development for teachers to help them understand the Essential Skills and their implications. However, across the state such efforts were sporadic and failed to reach significant numbers of teachers.
Since then, political processes have complicated the Arizona situation. The state superintendent of schools who was in office when the reform efforts were initiated did not stand for reelection in November 1994. Subsequently, the new state superintendent suspended plans for assessment. Revised standards and plans for assessment procedures have been adopted. The emphasis on assessment, rather than on curriculum and instruction, continues to be prominent.
METHOD
SELECTION
Participants in this study were K-12 mathematics teachers, elementary and middle school principals, and high school mathematics department chairs from 135 public and 42 private schools. Participating schools were chosen using a two-step procedure that employed stratified sampling by urban-rural location and high-low socioeconomic status (SES) levels.
Superintendents, then principals, of all selected schools were contacted to participate in the survey. Several declined for various reasons (e.g., involvement in other projects, curriculum revision). In particular, a number of urban districts chose not to participate because of pending lawsuits concerned with access to personnel files. Because of low participation rates by some SES and location groups, the investigators determined that generalizations for such groups would be limited or misleading. Thus, data were collapsed across SES and location.
Returns ranged from 35 to 45 percent for the first round and 50 to 60 percent for the second round. Responses for the two rounds were compared on 25 key variables. Significant between-round differences were found on only two items, suggesting that nonresponse error was not an important variable.
INSTRUMENT DEVELOPMENT
Questionnaire and interview instruments were used to gather information from teachers, principals, and department chairs. Investigators actively chose to use open-ended questions to avoid situations in which respondents could check every box that might appear on a closed-ended questionnaire. The investigators administered early versions of questionnaires and interview protocols to several groups in a pilot study. Based on the results, the instruments were revised.
PROCEDURES
Each participant received a personalized cover letter that outlined the study and its purposes, along with a stamped, self-addressed envelope. After approximately five weeks, second-round questionnaires were sent to individuals who had not responded initially. A request on the final page of the questionnaire solicited participation of interviewees. Teachers and administrators who were willing to provide additional information were asked to provide telephone numbers where they could be reached for interviews. From the pool of potential interviewees, the investigators selected teachers from urban and rural districts and, in elementary schools, a range of grade levels. Telephone interviews, conducted by trained interviewers, used detailed follow-up questions and actively sought fully developed responses.
Coding schemes were developed for responses to the open-ended questionnaire items. Investigators developed and critiqued the coding schemes; then they trained student assistants to code the questionnaire responses. High levels of interrater agreement, ranging between 93 and 96 percent, were attained. Subsequently, data were entered into computer files. Data were analyzed through frequency, cross-tabulation, and descriptive statistics from the SPSS procedures.
RESULTS
DESCRIPTION OF PARTICIPANTS
Two hundred eighty elementary teachers, most of them white females, responded to the questionnaire. Their undergraduate preparation for teaching averaged 9.5 credits in mathematics and 5 credits in mathematics methods; 10 percent or fewer held minors in mathematics. Approximately 50 percent of the teachers had a master's degree or 40 hours beyond their bachelor's degree. They averaged 12.97 years of teaching experience.
One hundred eighty-eight secondary teachers responded. Most were white; 45 percent were female. These teachers averaged 26.7 credits in mathematics and 7 credits in mathematics methods. Approximately 65 percent had a master's degree or 40 hours beyond their bachelor's degree. These teachers had an average of 12.64 years of teaching experience.
More teachers and administrators volunteered than could be interviewed, given time constraints and project staff. Thirty-seven teachers in grades K-8, 27 secondary mathematics teachers, 4 high school mathematics department chairs, and 16 elementary or middle school principals, generally representative of the sample, participated in interviews.
CURRICULUMS AND INSTRUCTIONAL PRACTICES
In this report, curriculum refers to what is taught to students, a deliberately open definition that acknowledges several levels based on their remoteness from students.(FN18) For example, an institutional curriculum, manifested in guides or scope-sequence charts, displays the district- or school-explicit curriculum. On the other hand, teachers prepare instructional curriculums that incorporate their own values and priorities concerning topics, time spent, method of delivery, and other considerations. Therefore, instructional curriculums differ in the degrees to which they reflect institutional curriculums.
Teachers' primary sources of instructional curriculum serve as important indices of their perspectives on mathematics content. Teachers who rely heavily on district guides, typically derived from Arizona Essential Skills for Mathematics or newly copyrighted texts, may embrace the reform version of content and teaching to a greater degree than teachers using other sources. For this reason, all teachers and department chairs were asked to respond to the open-ended questionnaire item, "What do you use to guide your mathematics curriculum?"
Participants typically listed between one and five sources; most listed more than one. Answers were classified according to categories that emerged in the analysis: textbooks, district or diocese guides, state guidelines, personal beliefs, specific mathematics programs, and other sources. The personal beliefs category encompasses related ideas including educators' experiences and perceptions of the needs of students. Particular programs include designated curriculum projects.
Table 1 reveals that results differed according to participant group. For example, whereas 44.3 percent of the elementary mathematics teachers named textbooks as a curriculum source, 57.1 percent of the mathematics chairs gave this response. Regardless of grade level, teachers use textbooks, district-diocese guides, and state guidelines as major curriculum sources. High school chairs also cited textbooks and state guidelines, but few named district guides as major sources.
TEACHERS' KNOWLEDGE OF CONTENT AND TEACHING STRATEGIES
Interviews provided additional information about teachers' evaluation of important content for students. These sessions also provided opportunities to question teachers about their instructional strategies.
Elementary school. Elementary teachers identified two or three topics that they believed to be essential for students at their grade levels. As expected, teachers differed in their choices of topics within and across grade levels. Despite this difference, half of the essential topics mentioned overall were clearly content, mostly arithmetic. About 12 percent of topics pertained to geometry and measurement. Teachers in kindergarten through 2nd grade named process skills, including problem solving, estimation, number sense, and patterns, much more often than did teachers in the other groups. Asked why these topics are important, most teachers indicated that the selected topics provide the foundation for future mathematics learning or for use in life outside of school.
All elementary teacher interviewees acknowledged that they taught fractions. To provide a focus for further inquiry, teachers in grades K-3 were asked, "What is the major idea you want students to understand about fractions as parts of wholes?" In grades 4-5, a similar question was applied to equivalent fractions/common denominators; and in grades 6-8, to the addition or subtraction of fractions with unlike denominators.
Analysis of interviewees' responses revealed that less than half (48 percent) of all teachers' responses incorporated the major mathematics concept. For example, many K-3 teachers' responses omitted any reference to "same-size parts," the major idea in fractional names for parts of wholes. This inadequacy in content understanding was substantiated in teachers' descriptions of how they routinely teach these topics.
Asked how they would teach the particular fraction topic, teacher interviewees' responses were vague. Teachers named manipulative materials suitable for instruction on the topics but provided few details about how these materials were to be used. Their expectation that students should experience actual hands-on use of materials or observe their use by others was not stated. How teachers help students to think about the mathematics ideas was also not communicated clearly. Teachers said that the materials were used to increase student understanding, but they offered no details even when interviewers pressed for answers.
Secondary school. High school teachers were interviewed about essential topics in Algebra I because this course furnishes basic understandings useful in both subsequent courses and the work force. Asked to name essential ideas, 64 percent of all the topics mentioned by the teachers related to solving equations, and 36 percent related to graphing. However, when these teachers were asked expressly to name the topics they teach in Algebra I, every teacher acknowledged teaching the same six equations-inequalities topics, and 64 percent listed graphing. Teachers' reasons for the importance of these topics ranged from preparation for subsequent mathematics study to recognition of the goal to increase students' critical thinking abilities as preparation for work outside of school.
Most high school teachers reported that they taught algebraic equations. They were asked, "What is the major idea you would want your students to know about the addition and multiplication properties of equality?" Analysis of the interviewees' responses revealed that about half (56 percent) of the teachers' responses incorporated the appropriate major idea of balancing equations as the underlying use of the properties. However, other teachers mentioned steps they taught for the solution of equations, an answer that may indicate a view of mathematics as facts and rules.
PROFESSIONAL DEVELOPMENT
During the 1993-94 school year only about 49 percent of the elementary teachers and 59 percent of the secondary teachers reported participation in any professional development related to mathematics content or methods. Those who participated actually spent little time in these programs--on average, one workday for elementary teachers and two workdays for secondary teachers. Still, both teachers and their supervisors rated the professional development programs as somewhat useful (ratings of 4.6 to 5.0 on a 7-point scale, where 7 is high).
Despite the brevity of professional development, at least 40 percent of the respondents reported making changes in their teaching strategies. These changes included the use of cooperative learning, improved questioning techniques, and hands-on learning activities, approaches recommended by the standards. Only about 10 percent of the elementary teachers and 6 percent of the secondary teachers reported changes in their knowledge of mathematics content, assessment, or students' mathematics needs. The extent to which this knowledge was applied in classrooms could not be determined.
Principals and high school chairs believed that teachers needed assistance with instruction and assessment to a greater extent than did teachers. Whereas 37.5 percent of principals and 51.7 percent of chairs indicated these teacher needs, only 18.2 percent of elementary teachers and 26.6 percent of secondary teachers viewed themselves as having these needs.
Elementary and middle school principals also indicated that their teachers had a fairly strong need for mathematics content (18.8 percent). However, only 4.8 percent of teachers saw mathematics content as a pressing need. Less than 10 percent of high school chairs and teachers perceived needs in content knowledge.
DISCUSSION
The results raise serious doubts that mathematics teachers in Arizona view mathematics content as dynamic or as process-oriented. Although less than half the teachers mention their use of state guidelines, district guidelines, or textbooks, a number of teachers are unaware of these curriculum sources.
Few of the elementary teachers interviewed reported knowledge of emphases on mathematical processes related to problem solving or reasoning. Only about half the teachers at elementary or secondary levels showed satisfactory understanding of the meaning of the mathematics concept targeted in their interviews. Teachers who report using the guidelines may use them to select mathematics topics, but they apparently attend little to what students are to understand or demonstrate about the topics.
On the surface, teachers' instructional strategies include activities similar to those included in the guidelines. Some use small groups and a variety of instructional materials, including manipulatives. However, the teachers seemed unable to articulate how these materials should be used and offered no hypotheses about how students learn with the materials.
Taken together, these results suggest that few Arizona teachers possess the perspectives on mathematics content and teaching described in the national standards and state guidelines. Teachers seem to be unaware of their deficiencies in content and instruction. Also, no teachers indicated a need for additional instruction about either.
Teachers in this study were experienced teachers, averaging more than 12 years of teaching experience. Many teachers completed their preparation for teaching before the initiation of the current wave of mathematics education reforms. Therefore, many have had little opportunity to learn about the reform efforts except, perhaps, in postbaccalaureate course work or other professional development programs.
Although half the elementary teachers and 65 percent of the secondary teachers had completed advanced college coursework, they may have had very little contact with knowledge of mathematics education reform. Only half the teachers in this study had any professional development in mathematics education during the 1993-94 school year. One day of professional development for elementary teachers and two days for secondary teachers, on average, appears to be insufficient.
As shown in this survey, teachers modify their teaching based on the professional development programs in which they are involved, regardless of the meagerness of such programs. Furthermore, teachers indicated their willingness to participate in additional professional development.
IMPLICATIONS FOR MATHEMATICS EDUCATION REFORM EFFORTS
In recent years many states have initiated efforts to reform mathematics offerings in schools through the use of the NCTM standards as the basis of state guidelines. Typically, these guidelines have been coupled with competency examinations to assess students' performance. In Arizona, this process has produced little reform in classrooms. The process simply ignores the fundamental idea that change resides within teachers, rather than in mandated curriculum or evaluation. The literature of educational innovation is clear that teachers are unlikely to change their behaviors without making changes in their beliefs and values.(FN19)
Results of this study suggest that state-district-diocese guidelines and updated textbooks provide insufficient incentive for teachers to change their practices. Assessment of student performance on new curriculums also is an anemic incentive, largely because both approaches are external to teachers. Missing in the process are changes in teachers' values and beliefs. Teachers who view mathematics as transmission of facts and rules teach this content to their students and are likely to continue to do so until they understand reasons to change.
Large-scale professional development initiatives should be undertaken to help Arizona teachers change their perspectives on mathematics content and teaching procedures. These initiatives must provide teachers with an understanding of the clear need for change, as well as time, resources, and encouragement to implement change in classrooms.
Added material
EVELYN SOWELL is Professor of Education at Arizona State University West, College of Education, P.O. Box 37100, Phoenix, AZ 85069-7100. RON ZAMBO is Assistant Professor of Mathematics Education at Arizona State University West, College of Education, P.O. Box 37100, Phoenix, AZ 85069-7100.
AUTHORS' NOTE: This article is based on a project supported, in part, by the Eisenhower Mathematics and Science Program. Opinions expressed are those of the authors and do not necessarily reflect those of the funding agency. The authors thank Saundra Bryn, Nancy Haas, and three anonymous reviewers for their comments on an earlier draft of this article.
Table 1. Teachers' Sources of Mathematics Curriculum Content (Percentages)
FOOTNOTES
1 National Council of Teachers of Mathematics, Curriculum and Evaluation Standards for School Mathematics (Reston, VA: NCTM, 1989); Diane Massell, "Setting Standards in Mathematics and Social Studies," Education and Urban Society 26 (February 1994): 118.
2 National Council of Teachers of Mathematics, Curriculum and Evaluation Standards for School Mathematics (Reston, VA: NCTM, 1989); National Council of Teachers of Mathematics, Professional Standards for Teaching Mathematics (Reston, VA: NCTM, 1991).
3 Larry Cuban, "Curriculum Stability and Change," in Handbook of Research on Curriculum, ed. Philip W. Jackson (New York: Macmillan, 1992), pp. 216-247.
4 Chris Pipho, "Centralizing Curriculum at the State Level," in The Politics of Curriculum Decision-Making: Issues in Centralizing the Curriculum, ed. M. Frances Klein (Albany, NY: State University of New York Press, 1991), pp. 67-97.
5 George F. Madaus and Thomas Kellaghan, "Curriculum Evaluation and Assessment," in Handbook of Research on Curriculum, ed. Philip W. Jackson (New York: Macmillan, 1992), pp. 137-147.
6 Since the early 1970s, researchers looking at the Concerns-Based Adoption Model have studied the change processes experienced by teachers and schools as they work on innovations. Among other findings, this research shows that change is a process and that individuals must change before organizations change. See Shirley Hord, Evaluating Educational Innovation (London: Croom Helm, 1987), and Gene E. Hall and Shirley Hord, Change in Schools: Facilitating the Process (Albany, NY: State University of New York Press, 1987). See also George Sharp, Curriculum Development as Re-education of the Teacher (New York: Teachers College, Columbia University, 1951).
7 Michael G. Fullan with Suzanne Stiegelbauer, The New Meaning of Educational Change (New York: Teachers College Press, 1991); Gene E. Hall and Shirley Hord, Change in Schools: Facilitating the Process (Albany, NY: State University of New York Press, 1987); Shirley Hord, Evaluating Educational Innovation (London: Croom Helm, 1987).
8 Michael G. Fullan with Suzanne Stiegelbauer, The New Meaning of Educational Change (New York: Teachers College Press, 1991).
9 Walter Doyle and Gerald A. Ponder, "The Practicality Ethic in Teacher Decision-Making," Interchange 8 (Summer 1977-78): 1-12.
10 William Bridges, Managing Transitions: Making the Most of Change (Reading, MA: Addison Wesley, 1991).
11 Michael T. Battista, "Teacher Beliefs and the Reform Movement in Mathematics Education," Phi Delta Kappan 75 (February 1994): 462-470.
12 William S. Bush, "Implementing the K-4 Mathematics Standards in Kentucky," Arithmetic Teacher 41 (November 1993): 166-169.
13 Ibid.; William S. Bush, "The Kentucky K-4 Mathematics Specialist Program," in Professional Development for Teachers of Mathematics, 1994 NCTM Yearbook, ed. Douglas B. Aichele and Arthur F. Coxford (Reston, VA: NCTM, 1994), pp. 246-254.
14 David D. Marsh and Allan R. Odden, "State-Initiated Curriculum Reform in Elementary School Mathematics and Science Programs" (paper presented at the annual meeting of the American Educational Research Association, Boston, April 1990).
15 Larry Cuban, "State-Powered Curricular Reform, Measurement-Driven Instruction," Phi Kappa Phi Journal 67 (Summer 1987); 23.
16 Arizona Essential Skills for Mathematics (Phoenix, AZ: Arizona Department of Education, 1992).
17 Ibid.
18 John I. Goodlad and Zhixin Su, "Organization of the Curriculum," in Handbook of Research on Curriculum, ed. Philip W. Jackson (New York: Macmillan, 1992), pp. 327-328.
19 Michael G. Fullan with Suzanne Stiegelbauer, The New Meaning of Educational Change (New York: Teachers College Press, 1991).
WBN: 9719602453005
Type in the content of your page here.